WEBVTT 00:00:00.040 --> 00:00:02.460 The following content is provided under a Creative 00:00:02.460 --> 00:00:03.870 Commons license. 00:00:03.870 --> 00:00:06.310 Your support will help MIT OpenCourseWare 00:00:06.310 --> 00:00:10.560 continue to offer high-quality educational resources for free. 00:00:10.560 --> 00:00:13.300 To make a donation or view additional materials 00:00:13.300 --> 00:00:17.116 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.116 --> 00:00:17.740 at ocw.mit.edu. 00:00:53.386 --> 00:00:54.300 PROFESSOR: Hi. 00:00:54.300 --> 00:00:58.050 Today we embark on a new phase of our course, 00:00:58.050 --> 00:01:01.130 signified by the beginning of block two. 00:01:01.130 --> 00:01:03.870 Where in block one, after indicating what 00:01:03.870 --> 00:01:07.570 mathematical structure was, in our quest 00:01:07.570 --> 00:01:11.000 to get as meaningfully as possible into the question 00:01:11.000 --> 00:01:12.930 of functions of several variables, 00:01:12.930 --> 00:01:15.690 we introduced the study of vectors. 00:01:15.690 --> 00:01:17.500 Vectors were important in their own right, 00:01:17.500 --> 00:01:20.970 so we paid some attention to that particular topic. 00:01:20.970 --> 00:01:24.420 And now we're at the next plateau, where one introduces 00:01:24.420 --> 00:01:27.580 the concept of functions. 00:01:27.580 --> 00:01:30.270 In other words, just like in ordinary arithmetic, 00:01:30.270 --> 00:01:33.260 after we get through with the arithmetic of constants, which 00:01:33.260 --> 00:01:35.600 is what you could basically call elementary school 00:01:35.600 --> 00:01:37.840 arithmetic-- you work with fixed numbers-- 00:01:37.840 --> 00:01:39.550 we then move into algebra. 00:01:39.550 --> 00:01:41.670 And from algebra we move into calculus. 00:01:41.670 --> 00:01:45.680 Sooner or later, we would like to study the calculus of vector 00:01:45.680 --> 00:01:47.130 functions and the like. 00:01:47.130 --> 00:01:49.850 And why not sooner? 00:01:49.850 --> 00:01:52.550 Which is what brings us to today's lesson. 00:01:52.550 --> 00:01:55.850 We are going to talk about a rather difficult mouthful, 00:01:55.850 --> 00:01:56.350 I guess. 00:01:56.350 --> 00:02:01.670 I call today's lecture "Vector Functions of Scalar Variables." 00:02:01.670 --> 00:02:04.470 And if that sounds like a tongue twister, 00:02:04.470 --> 00:02:07.950 let me point out that what has happened now is the following. 00:02:07.950 --> 00:02:10.479 When we did part one of this course, 00:02:10.479 --> 00:02:13.580 when we talked about functions and function machines, 00:02:13.580 --> 00:02:17.040 recall that both the input and the output of our machine 00:02:17.040 --> 00:02:20.352 were, by definition, scalars-- in other words, real numbers. 00:02:20.352 --> 00:02:21.810 Remember, we talked about functions 00:02:21.810 --> 00:02:23.990 of a single real variable. 00:02:23.990 --> 00:02:29.380 Now we have at our disposal both vectors and scalars. 00:02:29.380 --> 00:02:33.690 Consequently, the input can be either a vector or a scalar, 00:02:33.690 --> 00:02:37.580 and the output can be either a vector or a scalar. 00:02:37.580 --> 00:02:41.380 And this gives us an almost endless number 00:02:41.380 --> 00:02:44.690 of ways of combining these, where by "almost endless," 00:02:44.690 --> 00:02:45.890 I mean four. 00:02:45.890 --> 00:02:48.190 And the reason I say "almost endless" is this. 00:02:48.190 --> 00:02:49.340 There are only four ways. 00:02:49.340 --> 00:02:50.470 What are the four ways? 00:02:50.470 --> 00:02:52.100 Well, the four ways are what? 00:02:52.100 --> 00:02:54.120 Imagine that x is a scalar. 00:02:54.120 --> 00:02:56.080 In other words, the input is a scalar. 00:02:56.080 --> 00:02:58.400 Then the two possibilities are what? 00:02:58.400 --> 00:03:01.470 That the output is either a scalar or a vector. 00:03:01.470 --> 00:03:02.910 On the other hand, the input could 00:03:02.910 --> 00:03:06.810 have been a vector, in which case, there would have been 00:03:06.810 --> 00:03:08.510 two additional possibilities. 00:03:08.510 --> 00:03:11.500 Namely, the output would be either a scalar or a vector. 00:03:11.500 --> 00:03:14.890 Now the reason I call these four possibilities endless, 00:03:14.890 --> 00:03:17.630 or nearly endless, is the fact that by the time 00:03:17.630 --> 00:03:21.180 we get through discussing all four possible cases, 00:03:21.180 --> 00:03:23.440 the course will be essentially over. 00:03:23.440 --> 00:03:26.720 And to give you some hindsight as to what I mean, 00:03:26.720 --> 00:03:29.580 remember that we had a rather long course that 00:03:29.580 --> 00:03:31.470 was part one of this course. 00:03:31.470 --> 00:03:33.760 And notice that part one of this course 00:03:33.760 --> 00:03:38.040 was concerned only with that one case in four 00:03:38.040 --> 00:03:40.630 where both the input and the output 00:03:40.630 --> 00:03:43.480 happen to be real numbers, scalars. 00:03:43.480 --> 00:03:45.660 And it took us that long to develop the subject 00:03:45.660 --> 00:03:46.410 called what? 00:03:46.410 --> 00:03:49.900 Real functions of a single real variable-- input 00:03:49.900 --> 00:03:52.150 a scalar, output a scalar. 00:03:52.150 --> 00:03:57.610 Now the one that we've chosen for our next block, 00:03:57.610 --> 00:03:59.250 next combination that we've chosen, 00:03:59.250 --> 00:04:02.790 is where the input will be a scalar and the output 00:04:02.790 --> 00:04:05.980 will a vector. 00:04:05.980 --> 00:04:08.010 And in terms of physical examples, 00:04:08.010 --> 00:04:10.630 this can be made very, very meaningful. 00:04:10.630 --> 00:04:14.050 For example, one very common physical situation 00:04:14.050 --> 00:04:17.280 is that when we study force on an object-- force 00:04:17.280 --> 00:04:21.160 is obviously a vector-- but that the force on an object 00:04:21.160 --> 00:04:23.710 usually varies with time. 00:04:23.710 --> 00:04:26.330 But time happens to be a scalar. 00:04:26.330 --> 00:04:29.080 In other words, we might, for example, 00:04:29.080 --> 00:04:33.950 write that the vector F is a function of the scalar t. 00:04:33.950 --> 00:04:37.200 And rather than talk abstractly, let me make up 00:04:37.200 --> 00:04:39.350 a pseudo-physical situation. 00:04:39.350 --> 00:04:41.600 By pseudo-physical, I mean I haven't 00:04:41.600 --> 00:04:44.160 the vaguest notion where this formula would come up 00:04:44.160 --> 00:04:45.380 in real life. 00:04:45.380 --> 00:04:47.590 And I think that after you saw my performance 00:04:47.590 --> 00:04:51.000 on the explanation of work equals force times distance 00:04:51.000 --> 00:04:54.750 and why objects don't rise under the influence of friction 00:04:54.750 --> 00:04:56.160 and what have you, you believe me 00:04:56.160 --> 00:04:59.740 when I tell you I have no feel for these physical things. 00:04:59.740 --> 00:05:01.320 But I say that semi-jokingly. 00:05:01.320 --> 00:05:04.020 The main reason is that it's irrelevant what 00:05:04.020 --> 00:05:05.990 the physical application is. 00:05:05.990 --> 00:05:07.630 The important point is that each of you 00:05:07.630 --> 00:05:10.250 will find physical applications in your own way. 00:05:10.250 --> 00:05:13.240 All we really need is some generic problem 00:05:13.240 --> 00:05:16.630 that somehow signifies what all other problems look like. 00:05:16.630 --> 00:05:18.320 What I'm driving at here is, imagine 00:05:18.320 --> 00:05:20.790 that we have a force F, which varies 00:05:20.790 --> 00:05:24.710 with time, written in Cartesian coordinates as follows. 00:05:24.710 --> 00:05:29.710 The force is e to the minus t times i plus j. 00:05:29.710 --> 00:05:31.890 Notice that t is a scalar. 00:05:31.890 --> 00:05:36.380 As t varies, the scalar e to the minus t varies. 00:05:36.380 --> 00:05:39.110 But the scalar e to the minus t varying 00:05:39.110 --> 00:05:43.920 means that the vector e to the minus t i is changing, 00:05:43.920 --> 00:05:44.930 is varying. 00:05:44.930 --> 00:05:49.520 In other words, notice that e to the minus t i plus j 00:05:49.520 --> 00:05:53.870 is a variable vector, even though t is a scalar. 00:05:53.870 --> 00:05:57.300 For example, when t happens to be 0, what is the force? 00:05:57.300 --> 00:05:59.710 When t is 0, this would read what? 00:05:59.710 --> 00:06:03.550 The force is e to the minus 0 i plus j. 00:06:03.550 --> 00:06:05.890 That's the same as saying-- since e to the minus 0 00:06:05.890 --> 00:06:08.950 is 1-- the force, when the time is 0, 00:06:08.950 --> 00:06:11.890 is i plus j, which is a vector. 00:06:11.890 --> 00:06:14.540 You see, that vector changes as t changes, 00:06:14.540 --> 00:06:17.530 but t happens to be a scalar. 00:06:17.530 --> 00:06:21.030 Now here's the interesting point or an interesting point. 00:06:21.030 --> 00:06:23.310 I look at this expression, and I'm tempted 00:06:23.310 --> 00:06:25.490 to say something like this. 00:06:25.490 --> 00:06:31.310 I say for large values of t, F of t is approximately j. 00:06:31.310 --> 00:06:33.120 Now how did I arrive at that? 00:06:33.120 --> 00:06:34.890 Well what I said was, is I said, you know, 00:06:34.890 --> 00:06:39.040 e to the minus t gets very close to 0 as t gets large. 00:06:39.040 --> 00:06:44.500 As t gets large, therefore, this component tends to 0. 00:06:44.500 --> 00:06:48.970 This component stays constantly j-- or the component is 1, 00:06:48.970 --> 00:06:50.860 the vector is j. 00:06:50.860 --> 00:06:55.250 So that as t gets large, F of t behaves like j. 00:06:55.250 --> 00:06:58.930 And what I'm leading up to is that intuitively, 00:06:58.930 --> 00:07:01.040 I am now using the limit concept. 00:07:01.040 --> 00:07:02.560 I'm saying to myself, what? 00:07:02.560 --> 00:07:05.110 The limit of F of t as t approaches 00:07:05.110 --> 00:07:08.050 infinity et cetera is the limit of this sum. 00:07:08.050 --> 00:07:10.260 The limit of the sum is the sum of the limits. 00:07:10.260 --> 00:07:12.680 And I now take the limit term by term. 00:07:12.680 --> 00:07:16.870 And I sense that this limit is 0 and that this one is 1. 00:07:16.870 --> 00:07:18.920 Now you see, the whole point is this. 00:07:18.920 --> 00:07:23.830 What I would like to do is to study vector calculus. 00:07:23.830 --> 00:07:25.010 Meaning in this case, what? 00:07:25.010 --> 00:07:30.200 The calculus of vector functions of a scalar variable, 00:07:30.200 --> 00:07:32.180 of a scalar input. 00:07:32.180 --> 00:07:36.610 Now here's why structure was so important in our course. 00:07:36.610 --> 00:07:39.600 What I already know how to do is study limits 00:07:39.600 --> 00:07:43.320 in the case where both my input and my output were scalars. 00:07:43.320 --> 00:07:45.180 What I would like to be able to do 00:07:45.180 --> 00:07:49.790 is structurally inherit that entire system. 00:07:49.790 --> 00:07:51.150 Because it's a beautiful system. 00:07:51.150 --> 00:07:52.460 I understand it well. 00:07:52.460 --> 00:07:54.640 I have a whole bunch of theorems in that system. 00:07:54.640 --> 00:07:58.270 If I could only incorporate that verbatim into vector 00:07:58.270 --> 00:08:01.540 calculus, not only would I have a unifying thread, 00:08:01.540 --> 00:08:05.930 but I can save weeks of work not having to re-derive theorems 00:08:05.930 --> 00:08:09.339 for vectors which automatically have to be true because 00:08:09.339 --> 00:08:10.130 of their structure. 00:08:10.130 --> 00:08:12.980 Now that's, again, a difficult mouthful. 00:08:12.980 --> 00:08:15.770 This is written up voluminously in our notes. 00:08:15.770 --> 00:08:17.970 It's emphasized in the exercises, 00:08:17.970 --> 00:08:21.040 but I thought that I would try to show you a little bit live 00:08:21.040 --> 00:08:22.580 what this thing really means. 00:08:22.580 --> 00:08:24.330 Because I think that somehow or other, 00:08:24.330 --> 00:08:28.680 you have to hear these ideas, rather than just read them, 00:08:28.680 --> 00:08:30.640 to get the true feeling. 00:08:30.640 --> 00:08:33.919 What we do, for example, is we re-visit limits. 00:08:33.919 --> 00:08:36.200 And we write down the definition of limit 00:08:36.200 --> 00:08:39.409 just as it appeared in the scalar case. 00:08:39.409 --> 00:08:41.950 And let's start with an intuitive approach first, 00:08:41.950 --> 00:08:44.890 and later on, we'll get to the epsilon-delta approach. 00:08:44.890 --> 00:08:47.460 We said that the limit of f of x as x approaches 00:08:47.460 --> 00:08:53.090 a equals L means that f of x is near L 00:08:53.090 --> 00:08:55.690 provided that x is "near" a. 00:08:55.690 --> 00:08:58.930 And I've written the word "near" in quotation marks here 00:08:58.930 --> 00:09:02.632 to emphasize that we were using a geometrical phrase-- well, 00:09:02.632 --> 00:09:04.340 I guess you can't call one word a phrase, 00:09:04.340 --> 00:09:08.180 but a geometrical term-- to emphasize an arithmetic 00:09:08.180 --> 00:09:08.720 concept. 00:09:08.720 --> 00:09:13.530 Namely, to say that f of x was near L where f of x and L 00:09:13.530 --> 00:09:17.200 were numbers meant quite simply that what? 00:09:17.200 --> 00:09:20.710 The numerical value of f of x was very nearly equal 00:09:20.710 --> 00:09:24.110 to the numerical value of L. And that's the same as saying what? 00:09:24.110 --> 00:09:28.740 That the absolute value of the difference f of x minus L 00:09:28.740 --> 00:09:31.010 was very small. 00:09:31.010 --> 00:09:32.830 I just mention that, OK? 00:09:32.830 --> 00:09:37.110 Now what we would like to do is invent a definition of limit 00:09:37.110 --> 00:09:40.400 for a vector-valued function of a scalar variable. 00:09:40.400 --> 00:09:42.400 One way of doing this is of course 00:09:42.400 --> 00:09:45.240 to make up a completely brand new definition that has 00:09:45.240 --> 00:09:47.160 nothing to do with the past. 00:09:47.160 --> 00:09:49.380 But as in every field of human endeavor, 00:09:49.380 --> 00:09:54.480 one likes to plan the future and further sorties 00:09:54.480 --> 00:09:58.530 after one has modeled the successes of the past. 00:09:58.530 --> 00:10:01.210 So what we do is a very simple device 00:10:01.210 --> 00:10:02.980 and yet very, very powerful. 00:10:02.980 --> 00:10:07.340 We vectorize the definition that has already served us so well. 00:10:07.340 --> 00:10:09.740 What I mean by that is, I go back 00:10:09.740 --> 00:10:12.800 to the definition which I've written here and put in vectors 00:10:12.800 --> 00:10:14.260 in appropriate places. 00:10:14.260 --> 00:10:17.490 For example, we're dealing with a vector function, 00:10:17.490 --> 00:10:20.040 so f has to have an arrow over it. 00:10:20.040 --> 00:10:23.130 Moreover, since f of x is a vector, 00:10:23.130 --> 00:10:26.100 anything that it approaches as a limit by definition 00:10:26.100 --> 00:10:27.860 must also be a vector. 00:10:27.860 --> 00:10:31.490 So that means the L must also be arrowized. 00:10:31.490 --> 00:10:32.810 All right? 00:10:32.810 --> 00:10:34.710 Put an arrow over the L. 00:10:34.710 --> 00:10:36.660 On the other hand, the x and the a 00:10:36.660 --> 00:10:40.890 remain left alone because after all, they are just scalars. 00:10:40.890 --> 00:10:46.010 Remember again, x and a are inputs of our function machine. 00:10:46.010 --> 00:10:47.560 And in the particular investigation 00:10:47.560 --> 00:10:50.430 that we're making, our input happens to be a scalar. 00:10:50.430 --> 00:10:52.850 It's only the output that's a vector. 00:10:52.850 --> 00:10:55.630 So let's go back and, just as I say, arrowize 00:10:55.630 --> 00:10:59.230 or vectorize whatever has to be vectorized here. 00:10:59.230 --> 00:11:01.990 So I now have a new definition, intuitively. 00:11:01.990 --> 00:11:03.650 The limit of the vector function f 00:11:03.650 --> 00:11:08.680 of x as x approaches a is the vector L means that the vector 00:11:08.680 --> 00:11:12.725 f of x is near the vector L, provided that the scalar x 00:11:12.725 --> 00:11:14.720 is near the scalar a. 00:11:14.720 --> 00:11:18.220 Now the point is that the "x is near a" 00:11:18.220 --> 00:11:20.770 doesn't need any further interpretation because a 00:11:20.770 --> 00:11:22.510 and x are still scalars. 00:11:22.510 --> 00:11:24.770 The question that comes up now is twofold. 00:11:24.770 --> 00:11:29.070 First of all, does it make sense to say that f of x is near L? 00:11:29.070 --> 00:11:31.950 Does that even make sense when you're talking about vectors? 00:11:31.950 --> 00:11:35.120 The second thing is, if it does make sense, 00:11:35.120 --> 00:11:37.110 is it the meaning that we want it to have? 00:11:37.110 --> 00:11:39.970 Does it capture our intuitive feeling? 00:11:39.970 --> 00:11:43.550 Let's take the questions in order of appearance. 00:11:43.550 --> 00:11:47.300 First of all, what does it mean to say that f of x-- the vector 00:11:47.300 --> 00:11:49.690 f of x-- is near the vector L? 00:11:49.690 --> 00:11:51.970 In fact, let's generalize that. 00:11:51.970 --> 00:11:54.210 Given any two vectors A and B, what 00:11:54.210 --> 00:11:59.600 does it mean to say that the vector A is near the vector B? 00:11:59.600 --> 00:12:01.270 Now what my claim is, is this. 00:12:01.270 --> 00:12:04.504 Let's draw the vectors A and B as arrows in the plane here. 00:12:04.504 --> 00:12:06.920 Let's assume that they start at a common origin, which you 00:12:06.920 --> 00:12:08.850 can always assume, of course. 00:12:08.850 --> 00:12:13.640 To say that A is nearly equal to B somehow means what? 00:12:13.640 --> 00:12:17.520 That the vector A should be very nearly equal to the vector B. 00:12:17.520 --> 00:12:20.060 Well remember, if two vectors start 00:12:20.060 --> 00:12:22.120 at a common point, the only way that they 00:12:22.120 --> 00:12:25.410 can be equal is if their heads coincide. 00:12:25.410 --> 00:12:28.820 Consequently, with A and B starting at a common point, 00:12:28.820 --> 00:12:31.840 to say that A is nearly equal to B 00:12:31.840 --> 00:12:34.340 should mean that the distance between the head of A 00:12:34.340 --> 00:12:36.890 and the head of B should be small. 00:12:36.890 --> 00:12:39.530 But how do we state the distance between the head of A 00:12:39.530 --> 00:12:40.570 and the head of B? 00:12:40.570 --> 00:12:44.130 Notice that that has a very convenient numerical form. 00:12:44.130 --> 00:12:47.700 Namely, the vector that joins A to B 00:12:47.700 --> 00:12:50.400 can be written as either A minus B or B minus A, 00:12:50.400 --> 00:12:52.390 depending on what sense you put on this. 00:12:52.390 --> 00:12:53.720 We don't care about the sense. 00:12:53.720 --> 00:12:55.650 All we care about is the magnitude. 00:12:55.650 --> 00:13:00.000 Let's just call this length the magnitude of A minus B. 00:13:00.000 --> 00:13:01.230 And now we're in business. 00:13:01.230 --> 00:13:03.190 Namely, we say, lookit. 00:13:03.190 --> 00:13:07.800 To say that A is near B means that A is nearly equal to B, 00:13:07.800 --> 00:13:13.190 which in turn means that the magnitude of A minus B-- 00:13:13.190 --> 00:13:16.110 the magnitude of the vector-- what vector?-- 00:13:16.110 --> 00:13:19.290 the vector A minus the vector B-- is small. 00:13:19.290 --> 00:13:22.110 And keep in mind that I'm now using 'small' 00:13:22.110 --> 00:13:23.840 as I did arithmetically. 00:13:23.840 --> 00:13:26.530 Because even though A and B are vectors, 00:13:26.530 --> 00:13:31.000 and so also is A minus B, the magnitude of a vector 00:13:31.000 --> 00:13:32.320 is a number. 00:13:32.320 --> 00:13:34.940 Note this very important thing. 00:13:34.940 --> 00:13:36.580 Even when I'm dealing with vectors, 00:13:36.580 --> 00:13:40.580 as soon as I mention magnitude, I'm talking about a number. 00:13:40.580 --> 00:13:42.690 In other words then, I now define 00:13:42.690 --> 00:13:47.320 A to be near B to mean that the magnitude of the difference 00:13:47.320 --> 00:13:51.690 between A and B is small. 00:13:51.690 --> 00:13:54.400 And again, question two-- does that 00:13:54.400 --> 00:13:56.320 caption my intuitive feeling? 00:13:56.320 --> 00:13:57.710 And the answer is, yes, it does. 00:13:57.710 --> 00:14:03.500 I want A to be near B to mean that they nearly coincide. 00:14:03.500 --> 00:14:06.490 And that is the same as saying that the magnitude of A minus B 00:14:06.490 --> 00:14:07.380 is small. 00:14:07.380 --> 00:14:09.730 At any rate, having gone through this 00:14:09.730 --> 00:14:11.790 from a fairly intuitive point of view, 00:14:11.790 --> 00:14:15.720 let's now revisit limits more rigorously and rewrite 00:14:15.720 --> 00:14:19.414 our limit definition in terms of epsilons and deltas. 00:14:19.414 --> 00:14:21.830 And by the way, I hope this doesn't look like anything new 00:14:21.830 --> 00:14:24.060 to you, what I'm going to be talking about next. 00:14:24.060 --> 00:14:26.380 It's our old definition of limit that 00:14:26.380 --> 00:14:29.540 was the backbone of the entire part one of this course. 00:14:29.540 --> 00:14:31.960 I not only hope that it looks familiar to you, 00:14:31.960 --> 00:14:35.800 I hope that you read this thing almost subconsciously 00:14:35.800 --> 00:14:37.520 as second nature by now. 00:14:37.520 --> 00:14:38.620 But the statement is what? 00:14:38.620 --> 00:14:42.120 The limit of f of x as x approaches a equals L means, 00:14:42.120 --> 00:14:44.430 given any epsilon greater than 0, 00:14:44.430 --> 00:14:47.520 we can find a delta greater than 0 such 00:14:47.520 --> 00:14:52.040 that whenever the absolute value of x minus a is less than delta 00:14:52.040 --> 00:14:54.340 but greater than 0, the implication 00:14:54.340 --> 00:14:57.670 is that the absolute value of f of x minus L 00:14:57.670 --> 00:14:59.510 is less than epsilon. 00:14:59.510 --> 00:15:02.730 Now if I go back to my structural properties again, 00:15:02.730 --> 00:15:06.340 it seems to me that what my first approximation, at least-- 00:15:06.340 --> 00:15:10.150 for a rigorous definition to limits of vector functions-- 00:15:10.150 --> 00:15:12.330 should be to just, as I did before, 00:15:12.330 --> 00:15:14.930 read through this definition verbatim. 00:15:14.930 --> 00:15:17.890 Don't change a thing, but just vectorize 00:15:17.890 --> 00:15:19.970 what has to be vectorized. 00:15:19.970 --> 00:15:22.740 And because this definition that I'm going to give 00:15:22.740 --> 00:15:26.000 is so important, I elect to rewrite it. 00:15:26.000 --> 00:15:27.740 And what you're going to see next 00:15:27.740 --> 00:15:30.390 is nothing more than a carbon copy, 00:15:30.390 --> 00:15:34.570 so to speak, of this, only with appropriate arrows being 00:15:34.570 --> 00:15:35.490 put in. 00:15:35.490 --> 00:15:39.150 Namely, I am going to give, as my rigorous definition 00:15:39.150 --> 00:15:41.350 of the limit of f of x as x approaches 00:15:41.350 --> 00:15:44.920 a equals the vector L-- it's going to mean what? 00:15:44.920 --> 00:15:48.950 Given epsilon greater than 0, I can find delta greater 00:15:48.950 --> 00:15:53.240 than 0 such that whenever the absolute value of x minus a 00:15:53.240 --> 00:15:57.340 is less than delta but greater than 0, the implication is 00:15:57.340 --> 00:16:00.175 that the magnitude-- see I don't say absolute value now, 00:16:00.175 --> 00:16:02.050 because notice, I'm not dealing with numbers. 00:16:02.050 --> 00:16:06.390 These are vectors-- but the magnitude of f of x minus L 00:16:06.390 --> 00:16:09.366 must be less than epsilon. 00:16:09.366 --> 00:16:13.260 Now you see what I've done: I've obtained the second definition 00:16:13.260 --> 00:16:16.830 from the first simply by appropriate vectorization. 00:16:16.830 --> 00:16:20.870 What I have to make sure of is what? 00:16:20.870 --> 00:16:22.760 That this thing still makes sense. 00:16:22.760 --> 00:16:24.380 And notice that it does. 00:16:24.380 --> 00:16:26.950 Notice that what this says, in plain English is, what? 00:16:26.950 --> 00:16:34.200 That if x is sufficiently close to a-- what? 00:16:34.200 --> 00:16:36.920 The magnitude of the difference between these two vectors 00:16:36.920 --> 00:16:38.410 is as small as we wish. 00:16:38.410 --> 00:16:41.289 But we call that magnitude what? 00:16:41.289 --> 00:16:42.830 The distance between the two vectors. 00:16:42.830 --> 00:16:45.020 This says what? 00:16:45.020 --> 00:16:49.160 Correlating our formal language with our informal language, 00:16:49.160 --> 00:16:50.640 this simply says what? 00:16:50.640 --> 00:16:54.170 That we can make f of x as near to L 00:16:54.170 --> 00:16:58.640 as we wish just by picking x sufficiently close to a. 00:16:58.640 --> 00:17:03.240 So now what that tells us is that the new limit definition 00:17:03.240 --> 00:17:06.626 makes sense, just by mimicking the old definition. 00:17:06.626 --> 00:17:08.000 And what do we mean by mimicking? 00:17:08.000 --> 00:17:11.180 We mean vectorize the old definition, 00:17:11.180 --> 00:17:13.730 go through it word for word and put in vectors 00:17:13.730 --> 00:17:15.579 in appropriate places. 00:17:15.579 --> 00:17:17.670 Now let's see what this means structurally. 00:17:17.670 --> 00:17:21.200 That's the whole key to our particular lecture today: 00:17:21.200 --> 00:17:23.109 the structural value of this. 00:17:23.109 --> 00:17:26.380 For example, what were some of the building blocks 00:17:26.380 --> 00:17:29.860 that we based calculus of a single variable on? 00:17:29.860 --> 00:17:33.160 Remember, derivative was defined as a limit, 00:17:33.160 --> 00:17:35.890 and therefore all of our properties about derivatives 00:17:35.890 --> 00:17:37.660 followed from limit properties. 00:17:37.660 --> 00:17:42.830 Well, among the limit properties that we had for part one where, 00:17:42.830 --> 00:17:43.330 what? 00:17:43.330 --> 00:17:45.220 Input and output were scalars. 00:17:45.220 --> 00:17:49.810 The limit of the sum was equal to the sum of the limits. 00:17:49.810 --> 00:17:52.480 And the limit of a product is equal to the product 00:17:52.480 --> 00:17:53.610 the limits. 00:17:53.610 --> 00:17:55.690 And remember, in all of our proofs, 00:17:55.690 --> 00:17:58.900 we use these particular structural properties. 00:17:58.900 --> 00:17:59.400 You see? 00:17:59.400 --> 00:18:01.020 The next thing we would like to know-- 00:18:01.020 --> 00:18:03.300 see after all, these structural properties, 00:18:03.300 --> 00:18:04.800 even though they're called theorems, 00:18:04.800 --> 00:18:07.290 essentially, they become the rules of the game 00:18:07.290 --> 00:18:08.560 once we've proven them. 00:18:08.560 --> 00:18:10.430 In other words, once we've proved 00:18:10.430 --> 00:18:14.050 using epsilons and deltas that this happens to be true, 00:18:14.050 --> 00:18:18.030 notice that we never again use the fact that these are 00:18:18.030 --> 00:18:22.100 epsilons and deltas, that we have epsilons and deltas when 00:18:22.100 --> 00:18:24.870 we use this particular result. We just write it down. 00:18:24.870 --> 00:18:27.730 Because once we've proven it, we can always use it. 00:18:27.730 --> 00:18:29.720 In other words, what I'm really saying is, 00:18:29.720 --> 00:18:31.440 let me vectorize this. 00:18:35.392 --> 00:18:37.640 And I'll put a question mark now, because you see, 00:18:37.640 --> 00:18:39.450 I don't know if this is true for vectors. 00:18:39.450 --> 00:18:40.867 I don't know if the limit of a sum 00:18:40.867 --> 00:18:42.325 is the sum of the limits when we're 00:18:42.325 --> 00:18:44.360 dealing with vectors rather than with scalars. 00:18:44.360 --> 00:18:45.980 But notice what I've done. 00:18:45.980 --> 00:18:51.020 I have structurally plagiarized from my original definition. 00:18:51.020 --> 00:18:53.069 All I did was I vectorized it. 00:18:53.069 --> 00:18:54.610 You see, what I'm going to have to do 00:18:54.610 --> 00:18:57.050 is to check from my original definition 00:18:57.050 --> 00:18:59.562 whether these properties still hold true. 00:18:59.562 --> 00:19:01.270 I'm going to talk about that in a minute. 00:19:01.270 --> 00:19:03.380 Let me just continue on for a while. 00:19:03.380 --> 00:19:05.430 See, similarly over here, we talked 00:19:05.430 --> 00:19:07.350 about the limit of a product equaling 00:19:07.350 --> 00:19:08.780 the product of the limits. 00:19:08.780 --> 00:19:10.840 So again, we'd like to use results like that. 00:19:10.840 --> 00:19:12.590 And somebody says, well, let's vectorize. 00:19:14.684 --> 00:19:16.350 I want to show you what I mean by saying 00:19:16.350 --> 00:19:19.389 that after you vectorize, you have to be darn careful 00:19:19.389 --> 00:19:21.430 that you don't just go around saying, look at it, 00:19:21.430 --> 00:19:22.090 this is legal. 00:19:22.090 --> 00:19:24.080 I put arrows over everything. 00:19:24.080 --> 00:19:26.410 Sure it's legal, but you may not have 00:19:26.410 --> 00:19:28.240 anything that's worth keeping. 00:19:28.240 --> 00:19:29.820 Look at this expression. 00:19:29.820 --> 00:19:33.030 What are f and g in this case? 00:19:33.030 --> 00:19:36.090 The way I've written it, these happen to be vectors. 00:19:36.090 --> 00:19:37.280 See, they're both arrowized. 00:19:37.280 --> 00:19:37.780 Right? 00:19:37.780 --> 00:19:38.870 They're vectors. 00:19:38.870 --> 00:19:41.110 Have we talked about the ordinary product 00:19:41.110 --> 00:19:42.650 of two vector functions? 00:19:42.650 --> 00:19:43.860 And the answer is no. 00:19:43.860 --> 00:19:45.930 When we have two vector functions, 00:19:45.930 --> 00:19:48.710 we must either have a dot or a cross 00:19:48.710 --> 00:19:51.210 in here-- the dot product or the cross product. 00:19:51.210 --> 00:19:53.210 And if we're not going to put a dot in here, 00:19:53.210 --> 00:19:57.190 the best we can talk about is scalar multiplication. 00:19:57.190 --> 00:20:00.500 In other words, let me-- before you tend to memorize this, 00:20:00.500 --> 00:20:02.370 let me cross that out. 00:20:02.370 --> 00:20:04.350 Because this is ambiguous. 00:20:04.350 --> 00:20:05.610 It doesn't make sense. 00:20:05.610 --> 00:20:07.610 And what I am saying is that unfortunately-- not 00:20:07.610 --> 00:20:09.943 unfortunately, but if I want to be rigorous about this-- 00:20:09.943 --> 00:20:12.430 there happens to be three interpretations here. 00:20:12.430 --> 00:20:14.190 Namely, I can do what? 00:20:14.190 --> 00:20:16.790 I can think of f(x) as being a scalar 00:20:16.790 --> 00:20:19.180 and g(x) as being a vector. 00:20:19.180 --> 00:20:24.190 Or I can think of f(x) and g(x) as both being vector functions, 00:20:24.190 --> 00:20:26.360 but in one case having the dot product 00:20:26.360 --> 00:20:28.180 and in the other case, the cross product. 00:20:28.180 --> 00:20:32.120 In other words, to vectorize this particular equation, 00:20:32.120 --> 00:20:37.800 I guess what I have to do is write down 00:20:37.800 --> 00:20:39.630 these three possibilities. 00:20:39.630 --> 00:20:41.950 Namely, notice in this case, this is what? 00:20:41.950 --> 00:20:45.360 This is a scalar multiple of a vector function. 00:20:45.360 --> 00:20:48.270 Granted that the scalar multiple is a variable in this case, 00:20:48.270 --> 00:20:49.300 this still makes sense. 00:20:49.300 --> 00:20:54.170 Namely, for a fixed x-- for a fixed x, f of x is a constant, 00:20:54.170 --> 00:20:55.830 and g of x is a vector. 00:20:55.830 --> 00:20:59.630 A constant times a vector is a vector. 00:20:59.630 --> 00:21:04.390 For a fixed x, f and g here are fixed vectors. 00:21:04.390 --> 00:21:07.320 And I can talk about the dot product of two vectors. 00:21:07.320 --> 00:21:11.050 And for a fixed x over here, f of x and g of x 00:21:11.050 --> 00:21:13.850 are again fixed vectors, and consequently, I 00:21:13.850 --> 00:21:15.630 can talk about their cross product. 00:21:15.630 --> 00:21:17.900 And the question is, will these properties still 00:21:17.900 --> 00:21:20.930 be true when we're dealing with vector functions? 00:21:20.930 --> 00:21:22.920 The answer happens to be yes. 00:21:22.920 --> 00:21:25.260 But it's not yes automatically. 00:21:25.260 --> 00:21:28.970 In other words, what we're going to do in the notes is 00:21:28.970 --> 00:21:31.770 to mimic-- to prove these results for vectors, what we 00:21:31.770 --> 00:21:35.610 are going to do is to mimic every single proof 00:21:35.610 --> 00:21:38.020 that we gave in the scalar case, only 00:21:38.020 --> 00:21:41.450 replacing scalars by appropriate vectors 00:21:41.450 --> 00:21:43.720 whenever this is supposed to happen. 00:21:43.720 --> 00:21:46.180 The trouble is that certain results which 00:21:46.180 --> 00:21:48.880 were true for scalars may not automatically 00:21:48.880 --> 00:21:50.710 be true for vectors. 00:21:50.710 --> 00:21:52.520 Let me give you an example. 00:21:52.520 --> 00:21:54.780 Somehow or other to prove that the limit of a sum 00:21:54.780 --> 00:21:56.400 was equal to the sum of the limits, 00:21:56.400 --> 00:21:59.470 we used the property of absolute values 00:21:59.470 --> 00:22:02.600 that said that the absolute value of a sum was less than 00:22:02.600 --> 00:22:05.290 or equal to the sum of the absolute values. 00:22:05.290 --> 00:22:07.130 And we proved that for numbers. 00:22:07.130 --> 00:22:09.150 Suppose I now vectorize this. 00:22:14.750 --> 00:22:15.480 See? 00:22:15.480 --> 00:22:17.530 This now means magnitudes, right? 00:22:17.530 --> 00:22:20.310 And if a and b are any vectors in the plane, 00:22:20.310 --> 00:22:23.090 a and b no longer have to be in the same direction-- 00:22:23.090 --> 00:22:27.390 how do we know that vector addition has the same magnitude 00:22:27.390 --> 00:22:30.240 property that scalar addition has? 00:22:30.240 --> 00:22:33.010 See, can we be sure that this recipe is true? 00:22:33.010 --> 00:22:36.510 Well let's go and see what this recipe means. 00:22:36.510 --> 00:22:39.400 By the way, this does happen to be true in vector arithmetic. 00:22:39.400 --> 00:22:42.130 And it happens to be known as the triangle inequality. 00:22:42.130 --> 00:22:44.870 And the reason for that is, if I draw a triangle calling 00:22:44.870 --> 00:22:47.700 two of the sides a and b, respectively, 00:22:47.700 --> 00:22:51.830 and the third side c, we do know from elementary geometry 00:22:51.830 --> 00:22:56.030 that the sum of the lengths of two sides of a triangle 00:22:56.030 --> 00:22:58.900 is greater than or equal to the length of the third side 00:22:58.900 --> 00:23:00.040 of the triangle. 00:23:00.040 --> 00:23:03.400 What I'm driving at now is-- let me just vectorize this and show 00:23:03.400 --> 00:23:05.480 you something that I think is very cute here. 00:23:05.480 --> 00:23:08.910 If I put arrows here, this becomes the vector a, now; 00:23:08.910 --> 00:23:12.896 this becomes the vector b; this becomes the vector c. 00:23:12.896 --> 00:23:13.750 OK? 00:23:13.750 --> 00:23:16.590 But in terms of our definition of vector addition, 00:23:16.590 --> 00:23:20.040 noticed that c goes from the tail of a to the head of b. 00:23:20.040 --> 00:23:23.380 a and b are lined up properly for addition. 00:23:23.380 --> 00:23:27.400 So this is the vector a plus b. 00:23:27.400 --> 00:23:29.940 Now state the triangle inequality 00:23:29.940 --> 00:23:31.620 in terms of this picture. 00:23:31.620 --> 00:23:34.280 The triangle inequality says what? 00:23:34.280 --> 00:23:37.330 The third side of the triangle-- well what length is this? 00:23:37.330 --> 00:23:39.630 If the vector is a plus b, the length 00:23:39.630 --> 00:23:43.540 is the magnitude of a plus b-- must be less than 00:23:43.540 --> 00:23:46.720 or equal to the sum of the lengths of the other two sides. 00:23:46.720 --> 00:23:52.730 But those are just this. 00:23:52.730 --> 00:23:55.960 And that verifies the recipe that we want. 00:23:55.960 --> 00:23:57.910 In other words, it's going to turn out 00:23:57.910 --> 00:24:03.060 that every vector property that we need-- what 00:24:03.060 --> 00:24:05.690 do we mean by need-- that happened to be true for scalar 00:24:05.690 --> 00:24:09.220 calculus-- is going to be sufficient in vector calculus 00:24:09.220 --> 00:24:09.910 as well. 00:24:09.910 --> 00:24:13.570 For example, when we proved the product rule for scalars-- 00:24:13.570 --> 00:24:15.570 not the product rule, but the limit of a product 00:24:15.570 --> 00:24:17.780 was the product of the limits-- we used such things 00:24:17.780 --> 00:24:20.770 as the absolute value of a product 00:24:20.770 --> 00:24:23.420 was equal to the product of the absolute values. 00:24:23.420 --> 00:24:26.330 Notice, for example, in terms of case one over here, 00:24:26.330 --> 00:24:31.080 if I vectorize b and leave a as a scalar, the magnitude 00:24:31.080 --> 00:24:35.330 of a times the vector b is equal to the magnitude 00:24:35.330 --> 00:24:38.730 of a times the magnitude of b, just by the definition of what 00:24:38.730 --> 00:24:40.350 we meant by a scalar multiple. 00:24:40.350 --> 00:24:42.600 After all, a times b meant what? 00:24:42.600 --> 00:24:45.670 You just kept the direction of b constant, 00:24:45.670 --> 00:24:50.840 but multiplied b by the magnitude of a. 00:24:50.840 --> 00:24:51.540 Meaning what? 00:24:51.540 --> 00:24:54.250 By a, if was a positive; minus a, if a was negative, 00:24:54.250 --> 00:24:55.975 this is certainly true. 00:24:55.975 --> 00:24:57.850 By the way, there is a very important caution 00:24:57.850 --> 00:24:59.470 that I'd like to warn you about. 00:24:59.470 --> 00:25:01.760 And this is done in great detail in the notes. 00:25:01.760 --> 00:25:06.650 It's very important to point out that when I vectorize this 00:25:06.650 --> 00:25:08.820 in terms of a dot and a cross product, 00:25:08.820 --> 00:25:12.510 it is not true that the magnitude of a dot b 00:25:12.510 --> 00:25:16.950 is equal to the magnitude of a times the magnitude of b. 00:25:16.950 --> 00:25:18.450 You see, what I'm really saying here 00:25:18.450 --> 00:25:25.320 is, how was a dot b defined? a dot b, by definition, was what? 00:25:25.320 --> 00:25:29.110 It was the magnitude of a times the magnitude of b, 00:25:29.110 --> 00:25:32.540 times the cosine of the angle between a and b. 00:25:32.540 --> 00:25:36.709 Notice that if I take magnitudes here-- 00:25:36.709 --> 00:25:39.250 I want to keep this separate, so you can see what I'm talking 00:25:39.250 --> 00:25:41.300 about here-- 00:25:41.300 --> 00:25:41.800 Lookit. 00:25:41.800 --> 00:25:44.326 This factor here can be no bigger than 1. 00:25:44.326 --> 00:25:46.200 But in general, it's going to be less than 1. 00:25:46.200 --> 00:25:48.950 In other words, notice that the magnitude of a dot b 00:25:48.950 --> 00:25:52.410 is actually less than or equal to the magnitude of a times 00:25:52.410 --> 00:25:53.520 the magnitude of b. 00:25:53.520 --> 00:25:57.800 In fact, its equality holds only if this is a 0 degree 00:25:57.800 --> 00:25:59.960 angle or 180 degree angle. 00:25:59.960 --> 00:26:02.320 Because only then is the magnitude of the cosine 00:26:02.320 --> 00:26:03.560 equal to 1. 00:26:03.560 --> 00:26:06.057 Similar result holds for the cross product. 00:26:06.057 --> 00:26:07.640 The interesting point is-- and I'm not 00:26:07.640 --> 00:26:09.181 going to take the time to do it here, 00:26:09.181 --> 00:26:12.050 because I want this lecture to be basically an overview-- 00:26:12.050 --> 00:26:16.190 but the important point is that you don't need equality 00:26:16.190 --> 00:26:17.370 to prove our limit theorems. 00:26:17.370 --> 00:26:19.190 Remember, every one of our limit theorems 00:26:19.190 --> 00:26:21.470 was a string of inequalities. 00:26:21.470 --> 00:26:23.080 And actually-- and as I say, I'm going 00:26:23.080 --> 00:26:24.980 to do this in more detail in the notes-- 00:26:24.980 --> 00:26:28.460 the only result that we needed to prove theorems 00:26:28.460 --> 00:26:32.190 even in the part one section of our course 00:26:32.190 --> 00:26:36.100 was the fact that the less than or equal part be valid. 00:26:36.100 --> 00:26:38.830 The fact that it was equal when we were dealing with numbers 00:26:38.830 --> 00:26:40.450 was like frosting on the cake. 00:26:40.450 --> 00:26:43.910 We didn't need that strong a condition. 00:26:43.910 --> 00:26:46.190 The key overview that I want you to get 00:26:46.190 --> 00:26:48.320 from this present discussion is that, when 00:26:48.320 --> 00:26:51.750 we vectorize our definition of limit, 00:26:51.750 --> 00:26:55.680 that that new definition, having the same structure 00:26:55.680 --> 00:26:58.840 as the old-- meaning that all the properties of magnitudes 00:26:58.840 --> 00:27:01.590 of vectors that we need are carried over 00:27:01.590 --> 00:27:04.870 from absolute values of numbers, all previous limit 00:27:04.870 --> 00:27:07.550 theorems-- what do I mean by "all previous limit theorems"? 00:27:07.550 --> 00:27:11.080 I mean all the limit theorems of part one of this course 00:27:11.080 --> 00:27:13.330 are still valid. 00:27:13.330 --> 00:27:16.300 And with that in mind, I can now proceed 00:27:16.300 --> 00:27:18.600 to differential calculus. 00:27:18.600 --> 00:27:19.559 Namely what? 00:27:19.559 --> 00:27:21.100 I'm going to do the same thing again. 00:27:21.100 --> 00:27:24.410 I write down the definition of derivative as it 00:27:24.410 --> 00:27:26.710 was in part one of our course. 00:27:26.710 --> 00:27:31.440 And now what I do is I just vectorize everything in sight, 00:27:31.440 --> 00:27:33.390 provided it's supposed to be vectorized. 00:27:33.390 --> 00:27:36.370 Remember x and delta x are scalars here. 00:27:36.370 --> 00:27:38.590 f is the function. 00:27:38.590 --> 00:27:40.720 I just vectorize our definition. 00:27:40.720 --> 00:27:42.870 The first thing I have to do is to check 00:27:42.870 --> 00:27:45.350 to see whether the new expression makes sense. 00:27:45.350 --> 00:27:47.970 Notice that the numerator of my bracketed expression 00:27:47.970 --> 00:27:49.240 is now a vector. 00:27:49.240 --> 00:27:51.180 The denominator is a scalar. 00:27:51.180 --> 00:27:54.520 And a vector divided by a scalar makes perfectly good sense. 00:27:54.520 --> 00:27:57.590 In other words, this can be read as what? 00:27:57.590 --> 00:28:01.605 The scalar multiple 1 over delta x times the vector 00:28:01.605 --> 00:28:04.930 f of x plus delta x minus f of x. 00:28:04.930 --> 00:28:07.570 So this definition makes very good sense. 00:28:07.570 --> 00:28:10.676 It captures the meaning of average rate of change 00:28:10.676 --> 00:28:11.800 because you see, it's what? 00:28:11.800 --> 00:28:14.950 It's the total change in f over the change 00:28:14.950 --> 00:28:16.720 in x equal to delta x. 00:28:16.720 --> 00:28:18.430 So it's an average rate of change. 00:28:18.430 --> 00:28:21.000 Again, that's discussed in more detail in the text 00:28:21.000 --> 00:28:23.030 and in our notes, and in the exercises. 00:28:23.030 --> 00:28:26.460 But the point is that since every derivative property 00:28:26.460 --> 00:28:28.980 that we had in part one of our course 00:28:28.980 --> 00:28:31.700 followed from our limit properties, 00:28:31.700 --> 00:28:33.540 the fact that the limit properties are 00:28:33.540 --> 00:28:37.440 the same for vectors as they are for scalars now 00:28:37.440 --> 00:28:41.850 means that all derivative formulas are still valid. 00:28:41.850 --> 00:28:45.157 In other words, the product rule is still going to hold. 00:28:45.157 --> 00:28:46.740 The derivative of a sum is still going 00:28:46.740 --> 00:28:48.410 to be the sum of the derivatives. 00:28:48.410 --> 00:28:51.010 The derivative of a constant is still going to be 0. 00:28:51.010 --> 00:28:52.060 And keep this in mind. 00:28:52.060 --> 00:28:53.200 It's very important. 00:28:53.200 --> 00:28:56.460 I could re-derive every single one of these results 00:28:56.460 --> 00:28:59.650 from scratch as if scalar calculus had never 00:28:59.650 --> 00:29:04.820 been invented, just by using my basic epsilon-delta definition. 00:29:04.820 --> 00:29:08.190 The beauty of structure is, is that since the structures are 00:29:08.190 --> 00:29:10.570 alike-- you see, since they are played 00:29:10.570 --> 00:29:15.200 by the same rules of the game-- the theorems of vector calculus 00:29:15.200 --> 00:29:18.545 are going to be precisely those of scalar calculus. 00:29:18.545 --> 00:29:22.370 And I don't have to take the time to re-derive them all. 00:29:22.370 --> 00:29:24.740 As I say in the notes, I re-derive a few 00:29:24.740 --> 00:29:27.690 just so that you can get an idea of how this transliteration 00:29:27.690 --> 00:29:28.760 takes place. 00:29:28.760 --> 00:29:32.480 At any rate, let me close today's lesson with an example. 00:29:32.480 --> 00:29:35.290 Let's take motion in a plane. 00:29:35.290 --> 00:29:38.810 Let's suppose we have a particle moving along a curve C. 00:29:38.810 --> 00:29:40.540 And that the motion of the particle 00:29:40.540 --> 00:29:43.930 is given in parametric form, meaning we know both the x- 00:29:43.930 --> 00:29:47.870 and the y-coordinates of the particle at any time t, 00:29:47.870 --> 00:29:49.890 as functions of t. 00:29:49.890 --> 00:29:53.110 Notice, by the way, that what requires 00:29:53.110 --> 00:29:57.620 two equations in scalar form can be written as a single vector 00:29:57.620 --> 00:30:01.880 equation, namely, at time t-- let's say at time t, 00:30:01.880 --> 00:30:03.790 the particle was over here. 00:30:03.790 --> 00:30:08.440 Notice that if we now let r be the vector from the origin 00:30:08.440 --> 00:30:13.200 to the position of the particle that r is a vector, right? 00:30:13.200 --> 00:30:16.530 Because to specify r, I not only have to give its length, 00:30:16.530 --> 00:30:17.970 I have to give its direction. 00:30:17.970 --> 00:30:22.601 r is a vector which varies with our scalar t. 00:30:22.601 --> 00:30:23.100 All right? 00:30:23.100 --> 00:30:26.400 So r is a vector function of the scalar t. 00:30:26.400 --> 00:30:29.880 Now what is the i component of r? 00:30:29.880 --> 00:30:32.710 Well, the i component is x. 00:30:32.710 --> 00:30:34.920 And the j component is y. 00:30:34.920 --> 00:30:40.160 Notice that the pair of simultaneous parametric scalar 00:30:40.160 --> 00:30:43.820 equations x equals x of t, y equals y of t 00:30:43.820 --> 00:30:48.230 can be written as the single vector equation r of t 00:30:48.230 --> 00:30:54.040 equals x of t i plus y of t j, where x of t and y of t 00:30:54.040 --> 00:30:57.580 are scalar functions of the scalar variable t. 00:30:57.580 --> 00:30:59.690 OK? 00:30:59.690 --> 00:31:01.820 Now the question comes up, wouldn't it 00:31:01.820 --> 00:31:04.380 be nice to just take dr/dt here? 00:31:04.380 --> 00:31:08.620 Well, I mean, that's about as motivated as you can get. 00:31:08.620 --> 00:31:09.180 Meaning what? 00:31:09.180 --> 00:31:10.138 Let's see what happens. 00:31:10.138 --> 00:31:11.550 I know how to differentiate now. 00:31:11.550 --> 00:31:14.500 I'm going to use the fact that the derivative of a product 00:31:14.500 --> 00:31:16.530 of a sum is a sum of the derivatives, 00:31:16.530 --> 00:31:18.290 that i and j are constants. 00:31:18.290 --> 00:31:21.310 And a constant times a function, to differentiate that, 00:31:21.310 --> 00:31:24.260 you skip over the constant and differentiate the function. 00:31:24.260 --> 00:31:27.320 x of t and y of t are scalar functions, 00:31:27.320 --> 00:31:30.050 and I already know how to differentiate scalar functions. 00:31:30.050 --> 00:31:34.030 So all I'm going to assume now is that x of t and y of t 00:31:34.030 --> 00:31:37.420 are differentiable scalar functions, which means now 00:31:37.420 --> 00:31:39.830 that instead of just talking about motion in a plane, 00:31:39.830 --> 00:31:41.980 I'm assuming that the motion is smooth. 00:31:41.980 --> 00:31:45.380 At any rate, I now differentiate term by term. 00:31:45.380 --> 00:31:46.769 And look what I get. 00:31:46.769 --> 00:31:47.310 This is what? 00:31:47.310 --> 00:31:51.730 It's dx/dt times i plus dy/dt times j. 00:31:51.730 --> 00:31:55.120 In other words, the dr/dt is this particular vector. 00:31:55.120 --> 00:31:57.850 And this particular vector is fascinating. 00:31:57.850 --> 00:31:59.380 Why is it fascinating? 00:31:59.380 --> 00:32:02.700 Well, for one thing, let's compute its magnitude. 00:32:02.700 --> 00:32:06.300 The magnitude of any vector in i and j components is, what? 00:32:06.300 --> 00:32:09.510 The square root of the sum of the squares of the components. 00:32:09.510 --> 00:32:11.010 In this case, that's the square root 00:32:11.010 --> 00:32:16.350 of dx/dt squared plus dy/dt squared. 00:32:16.350 --> 00:32:18.360 But remember from part one of our course, 00:32:18.360 --> 00:32:23.250 this is exactly the magnitude of ds/dt where s is arc length. 00:32:23.250 --> 00:32:25.900 Remember I put the absolute value sign in here 00:32:25.900 --> 00:32:29.250 because we always take the positive square root. 00:32:29.250 --> 00:32:30.670 We're assuming that arc length is 00:32:30.670 --> 00:32:35.290 traversed in a given direction at a particular time t. 00:32:35.290 --> 00:32:36.920 But what is ds/dt? 00:32:36.920 --> 00:32:40.020 It's the change in arc length with respect to time. 00:32:40.020 --> 00:32:43.860 And that's precisely what we mean by speed along the curve. 00:32:43.860 --> 00:32:48.210 On the other hand, if we look at the slope of dr/dt, it's what? 00:32:48.210 --> 00:32:50.230 It's the slope-- it's determined by what? 00:32:50.230 --> 00:32:53.180 You take the j component, which is dy/dt, 00:32:53.180 --> 00:32:56.480 divided by the i component, which is dx/dt. 00:32:56.480 --> 00:32:59.610 By the chain rule, we know that that's dy/dx. 00:32:59.610 --> 00:33:03.400 Therefore, the vector dr/dt has its magnitude 00:33:03.400 --> 00:33:05.640 equal to the speed along the curve. 00:33:05.640 --> 00:33:08.480 And its direction is tangential to the curve. 00:33:08.480 --> 00:33:10.800 And what better motivation than that 00:33:10.800 --> 00:33:13.280 to define a velocity vector? 00:33:13.280 --> 00:33:15.670 After all, we've already done that in elementary physics. 00:33:15.670 --> 00:33:18.130 In elementary physics, what was the velocity vector 00:33:18.130 --> 00:33:19.320 defined to be? 00:33:19.320 --> 00:33:21.440 At a given point, it was the vector 00:33:21.440 --> 00:33:24.040 whose direction to the curve-- whose direction 00:33:24.040 --> 00:33:26.670 was tangential to the path at that point 00:33:26.670 --> 00:33:29.120 and whose magnitude was numerically 00:33:29.120 --> 00:33:31.150 equal to the speed along the curve 00:33:31.150 --> 00:33:32.860 that the particle had at that point. 00:33:32.860 --> 00:33:35.340 And that's precisely what dr/dt has. 00:33:35.340 --> 00:33:37.090 In other words, what we have done 00:33:37.090 --> 00:33:40.890 is given a self-contained mathematical derivation 00:33:40.890 --> 00:33:44.260 as to why the derivative of the position vector r, 00:33:44.260 --> 00:33:46.690 with respect to time, should physically 00:33:46.690 --> 00:33:49.060 be called the velocity vector. 00:33:49.060 --> 00:33:52.260 And then again, analogously to ordinary physics, 00:33:52.260 --> 00:33:56.320 if v is the velocity vector, we define the acceleration vector 00:33:56.320 --> 00:33:58.910 to be the derivative of the velocity vector with respect 00:33:58.910 --> 00:33:59.860 to time. 00:33:59.860 --> 00:34:01.920 Now I was originally going to close over here, 00:34:01.920 --> 00:34:03.570 but my feeling is that this seems 00:34:03.570 --> 00:34:05.580 a little bit abstract to you, so maybe we 00:34:05.580 --> 00:34:07.070 should take a couple more minutes 00:34:07.070 --> 00:34:09.670 and do a specific problem. 00:34:09.670 --> 00:34:12.960 Let's take the particle that moves along the path whose 00:34:12.960 --> 00:34:16.850 parametric equations are y equals t cubed plus 1, 00:34:16.850 --> 00:34:18.570 and x equals t squared. 00:34:18.570 --> 00:34:21.595 In other words, notice that for a given value of t-- t 00:34:21.595 --> 00:34:24.000 is a scalar-- for a given value of t, 00:34:24.000 --> 00:34:26.090 I can figure out where the particle is 00:34:26.090 --> 00:34:27.750 at any time along the curve. 00:34:27.750 --> 00:34:34.690 For example, when the time is 2, when t is 2, x is 4, y is 9. 00:34:34.690 --> 00:34:37.710 So at t equals 2, the particle is at the point 4 comma 00:34:37.710 --> 00:34:39.520 9, et cetera. 00:34:39.520 --> 00:34:41.219 Notice that to understand this problem, 00:34:41.219 --> 00:34:42.900 one does not need vectors. 00:34:42.900 --> 00:34:48.560 But if one knows vectors, one introduces the radius vector r. 00:34:48.560 --> 00:34:49.949 What is the radius vector? 00:34:49.949 --> 00:34:52.880 Its i component is the value of x, 00:34:52.880 --> 00:34:55.480 and its j component is the value of y. 00:34:55.480 --> 00:34:58.470 So the radius vector r is t squared i 00:34:58.470 --> 00:35:00.980 plus t cubed plus 1 j. 00:35:00.980 --> 00:35:04.230 Now, by these results, I can very quickly 00:35:04.230 --> 00:35:07.195 compute both the velocity and the acceleration of my particle 00:35:07.195 --> 00:35:08.410 at any time. 00:35:08.410 --> 00:35:10.670 Namely, to find the velocity, I just 00:35:10.670 --> 00:35:12.621 differentiate this with respect to t. 00:35:12.621 --> 00:35:13.870 This is going to give me what? 00:35:13.870 --> 00:35:16.130 The derivative of t squared is 2t. 00:35:16.130 --> 00:35:20.730 The derivative of t cubed plus 1 is 3 t squared. 00:35:20.730 --> 00:35:26.460 So my velocity vector is just 2t*i plus 3 t squared j. 00:35:26.460 --> 00:35:28.500 Now to get the acceleration vector, 00:35:28.500 --> 00:35:30.940 I simply have to differentiate the velocity vector 00:35:30.940 --> 00:35:32.980 with respect to time. 00:35:32.980 --> 00:35:34.500 And I get what? 00:35:34.500 --> 00:35:37.130 2i plus 6t*j. 00:35:37.130 --> 00:35:40.200 And I now have the acceleration, the velocity, 00:35:40.200 --> 00:35:44.195 and, by the way, the position of my particle at any time t. 00:35:44.195 --> 00:35:45.570 In particular, since I've already 00:35:45.570 --> 00:35:47.525 computed that the particle is at the point 00:35:47.525 --> 00:35:50.500 4 comma 9 when t is 2, let's carry 00:35:50.500 --> 00:35:53.370 through the rest of our investigation when t is 2. 00:35:53.370 --> 00:35:56.980 When t is 2, notice that in vector form, 00:35:56.980 --> 00:36:05.280 r is equal to 4i plus 9j; v is equal to 4i plus 12j; 00:36:05.280 --> 00:36:12.100 and a is equal to 2i plus 12j. 00:36:12.100 --> 00:36:15.030 Pictorially-- and by the way, to show you how to get this, 00:36:15.030 --> 00:36:17.630 all I'm saying here is that if we come back 00:36:17.630 --> 00:36:22.900 here recalling that t cubed is t squared to the 3/2 power-- 00:36:22.900 --> 00:36:27.010 this simply says that y is equal to x to the 3/2 plus 1. 00:36:27.010 --> 00:36:28.756 So I drew the path in just so that you 00:36:28.756 --> 00:36:30.890 can get an idea of what's going on over here. 00:36:30.890 --> 00:36:32.690 All I'm saying is when somebody says, 00:36:32.690 --> 00:36:36.090 where is the particle at time t equals 2, it's at the point 00:36:36.090 --> 00:36:37.620 4 comma 9. 00:36:37.620 --> 00:36:39.320 And notice the correlation again. 00:36:39.320 --> 00:36:43.650 The point 4 comma 9 is the point at which the vector 4i 00:36:43.650 --> 00:36:45.700 plus 9j terminates. 00:36:45.700 --> 00:36:49.390 Because you see, that vector originates at (0, 0). 00:36:49.390 --> 00:36:52.630 I can compute the velocity vector-- namely what? 00:36:52.630 --> 00:36:57.070 The velocity vector has its slope equal to 3-- 12 over 4. 00:36:57.070 --> 00:36:59.280 And its magnitude is the square root of 4 00:36:59.280 --> 00:37:01.080 squared plus 12 squared. 00:37:01.080 --> 00:37:04.380 That's the square root of 160, which is roughly 12 and a half. 00:37:04.380 --> 00:37:07.300 I can now draw that velocity vector to scale. 00:37:07.300 --> 00:37:10.530 In a similar way, the slope of the acceleration vector 00:37:10.530 --> 00:37:13.740 at that point is 12 over 2, which is 6. 00:37:13.740 --> 00:37:15.640 So the slope is 6. 00:37:15.640 --> 00:37:17.720 That's a pretty steep line here. 00:37:17.720 --> 00:37:19.110 And the magnitude is what? 00:37:19.110 --> 00:37:22.590 The square root of 4 plus 144-- the square root 00:37:22.590 --> 00:37:25.400 of 148, which I just call approximately 12, 00:37:25.400 --> 00:37:26.730 to give you a rough idea. 00:37:26.730 --> 00:37:29.260 I can draw this into scale as well. 00:37:29.260 --> 00:37:33.950 In other words, in just one short overview lesson, 00:37:33.950 --> 00:37:36.210 notice that we have introduced what 00:37:36.210 --> 00:37:39.440 we mean by vector functions, what we mean by limits. 00:37:39.440 --> 00:37:42.850 We've inherited the entire structure of part one. 00:37:42.850 --> 00:37:45.500 I can now introduce a position vector, 00:37:45.500 --> 00:37:47.320 differentiate it with respect to time 00:37:47.320 --> 00:37:49.640 to find velocity and acceleration, 00:37:49.640 --> 00:37:52.620 and I'm in business, being able to solve 00:37:52.620 --> 00:37:55.270 vector problems in the plane-- kinematics problems, 00:37:55.270 --> 00:37:57.270 if you will. 00:37:57.270 --> 00:38:00.160 We're going to continue on in this vein for the remainder 00:38:00.160 --> 00:38:00.820 of this block. 00:38:00.820 --> 00:38:03.300 We have other coordinate systems to talk about. 00:38:03.300 --> 00:38:05.500 Next time, I'm going to talk about 00:38:05.500 --> 00:38:09.690 tangential and normal components of vectors and the like. 00:38:09.690 --> 00:38:12.860 But all the time, the theory that we're talking about 00:38:12.860 --> 00:38:13.940 is the same. 00:38:13.940 --> 00:38:17.470 Namely, we are given motion in a plane. 00:38:17.470 --> 00:38:20.250 And we can now, in terms of vector calculus, 00:38:20.250 --> 00:38:23.570 study that motion very, very effectively. 00:38:23.570 --> 00:38:25.840 At any rate, until next time, goodbye. 00:38:29.230 --> 00:38:31.600 Funding for the publication of this video 00:38:31.600 --> 00:38:36.480 was provided by the Gabriella and Paul Rosenbaum Foundation. 00:38:36.480 --> 00:38:40.650 Help OCW continue to provide free and open access to MIT 00:38:40.650 --> 00:38:45.070 courses by making a donation at ocw.mit.edu/donate.